We provide a lattice regularization of all topological defects in minimal models CFTs using RSOS and anyonic spin chains. For defects of type
(1,s), we connect our result with the "topological symmetry" initially identified in Fibonacci anyons [Phys. Rev. Lett. 98, 160409 (2007)], and the center of the affine Temperley-Lieb algebra discussed in [1811.02551]. We show that the topological nature of the defects is exact on the lattice as well. Our defects of type
(r,1), in contrast, are only topological in the continuum limit. Identifications are obtained by a mix of algebraic and Bethe-ansatz techniques. Most of our discussion is framed in a Hamiltonian (or transfer matrix) formalism, and direct and crossed channel are both discussed in detail. For defects of type
(1,s), we also show how to implement their fusion, which turns out to reproduce the tensor product of the underlying monoidal category used to build the anyonic chain.
Abstract The equivalent of fusion in boundary conformal field theory (CFT) can be realized quite simply in the context of lattice models by essentially glueing two open spin chains. This has led to many developments, in particular in the context of chiral logarithmic CFT. We consider in this paper a possible generalization of the idea to the case of bulk conformal field theory. This is of course considerably more difficult, since there is no obvious way of merging two closed spin chains into a big one. In an earlier paper, two of us had proposed a “topological” way of performing this operation in the case of models based on the affine Temperley-Lieb (ATL) algebra, by exploiting the associated braid group representation and skein relations. In the present work, we establish — using, in particular, Frobenius reciprocity — the resulting fusion rules for standard modules of ATL in the generic as well as partially degenerate cases. These fusion rules have a simple interpretation in the continuum limit. However, unlike in the chiral case this interpretation does not match the usual fusion in non-chiral CFTs. Rather, it corresponds to the glueing of the right moving component of one conformal field with the left moving component of the other.
We discuss in this paper the lattice discretizations of all topological defect lines (TDLs) for diagonal, minimal CFTs, using integrable restricted solid-on-solid (RSOS) models. For these CFTs, the TDLs can be labeled by the Kac labels. In the case of
(1,s) TDLs, lines that are exactly topological on the lattice can be obtained using the centralizer of the underlying Temperley-Lieb algebra, all the other lines become topological in the continuum limit only. Our general construction relies on insertions of rows/columns of faces with modified spectral parameters, and can therefore be studied using integrability techniques. We determine the regions of spectral parameters realizing the different
(r,s) TDLs, and in particular calculate analytically all the associated eigenvalues (and degeneracy factors). We also show how fusion of TDLs can be obtained from fusion hierarchies in the algebraic approach to the Bethe-ansatz. All our results are checked numerically in detail for several minimal CFTs.